Methods are known and used for reconstructing images from data sets obtained by undersampling, and use a technique known as Compressed Sensing (CS), having the purpose of reducing the amount of data to be sampled during acquisition.
The basic principle of CS consists in measuring a few critical coefficients of a compressible signal and then reconstructing it by an iterative optimization process.
This reconstruction procedure is based on the solution of the following constrained optimization problem;min∥Ψm∥1s.t.∥Fum−y∥2<ε,  (1)
where m is the reconstructed image, Ψ is the transform in a domain, where the result of the transform is sparse, preferably the Wavelet transform, Fu is the Fourier transform with the same undersampling as the acquired k-space y. This procedure tends to an image m whose Wavelet transform is sparse and whose k-space is coherent with the acquired k-space (fidelity term).
The term of fidelity to the acquired image data is a rank-2 minimization, i.e. sum of the squares of the moduli, the difference between the image data obtained at each iteration step and the actually acquired image data.
A further term relating to a further space in which the image is mapped in sparse mode is the term of the finite differences of the image, usually known as Total Variation (TV), whose introduction causes the equation (1) to become:min∥Ψm∥1+αTV(m)s.t.∥Fum−y∥2<ε,  (2)
Using Lagrange multipliers, the constraints may be introduced in the function to be minimized, which becomes:f=∥Fum−y∥2+λ1∥Ψm∥1+λ2TV(m),  (3)
where the first term is the term of fidelity to the acquired image data, the second term is the term of image data sparsity in a predetermined domain, preferably in the Wavelet domain, and the third term is the term of finite differences of the image, or Total Variation.
Methods are known which use CS in MRI and ultrasound imaging.
Nevertheless, prior art methods are directed to undersampled acquisition and reconstruction either of individual images from individual sets of image data, or dynamic time-based images.
However, in many situations, multiple sets of image data are acquired, for a number of reasons.
In these situations, prior art methods can only reduce acquisition time by undersampling, but no use is made of redundant information for artifact correction, and there is no change of the algorithm for incorporating therein a prior knowledge of such artifacts to be removed.